Edited on 12/24/2014 to reflect release of the 3 core rule-books.
Starter Set
5th edition's Starter Set is decent to give you a feel for the basic rules, and outside players are not necessary. It is designed with beginners in mind, and is thus a good intro to the game. The starter set has an adventure in it that goes from level 1 to 5, while covering a lot of ground from a role-playing perspective as well as a combat perspective.
It's 20 dollars, but there's a high likelihood that most things outside of using the basic rules to create a character and joining the adventure league as a player are going to cost you more.
It does give you enough to work with to create your own campaigns as well, as the monsters in the appendix are varied enough for some play-room.
Extant Groups
This is particularly advantageous if you don't have a lot of people in your current group and want to get going. If you just combine two groups, you might make the new group too big to manage reasonably. If there's only a couple of you it is easy and would likely be a great intro. If you have a single player you can draw in to play or DM that knows a lot about the game, it can certainly compensate on either side.
Basic Rules
The basic rules are available on the official D&D website. They cover character creation for one of each type of fighter, cleric, rogue, and wizard as well as an assortment of backgrounds. The basic rules are free, and include monsters and magic items in the Dungeon Master documents. They won't tell your DM how to prepare a game outside of managing encounters, but it is enough to get you moving for free if your DM is willing to put in a little work.
Core Rules
As of the time of this update, the three core rule-books are available for purchase at online retailers and local game/hobby-stores.
The Player's Handbook (PHB) covers all of character creation, including far more options than the basic rules and documentation of the rules necessary to play that is more in depth than the basic rules (except in combat, which was covered very well in the basic rules). It is about the price of a new computer game (40-60 USD after-tax depending on where you get them). A game benefits highly from having this book, and most players end up getting one for themselves at some point.
The Monster Manual (MM) and Dungeon Master's Guide (DMG) are more for the DM. The former includes a huge assortment of monsters, beasts, and NPC templates ready for use. The latter is primarily about customizing your campaign, tuning and tweaking it for the way you want to play. The DMG also covers magic items, random dungeon generators, and a host of other resources that can be useful to someone running a game to speed up preparation and allow for customization. Typically only one of each of these books is necessary in an entire gaming group, but with the basic rules for free, the investment can wait until you are ready for it. Each of these two books costs the same as the PHB. Some groups pitch in together to split the cost for these two books, just make sure there won't be problems with book ownership after it's all done.
Summation
If you are in a hurry to try it within the next 3 or 4 weeks, then by all means pick up the Starter Set, or find someone at your local game store that has it and wants to run a game. Ultimately, the decision on how to approach this is up to you. There is no wrong path to start the game.
From the PHB (p. 196):
Critical Hits
When you score a critical hit, you get to roll extra dice
for the attack’s damage against the target. Roll all of the
attack’s damage dice twice and add them together. Then
add any relevant modifiers as normal. To speed up play,
you can roll all the damage dice at once.
For example, if you score a critical hit with a dagger,
roll 2d4 for the damage, rather than 1d4, and then add
your relevant ability modifier. If the attack involves other
damage dice, such as from the rogue’s Sneak Attack
feature, you roll those dice twice as well.
The example given answers your first question perfectly.
For the spider, the secondary damage is from poison, so is not inherently part of the attack. This can be seen by the fact that the damage doesn't depend on the attack roll, but a separate saving throw.
Critical hits represent hitting a vulnerable area. With poison, it doesn't really matter where you are hit.
Best Answer
Yes.
The phrasing for critical hits in 5e is the general attack. This counts for anything with an attack roll, a monster attack, an adventurer attack, a generic NPC attack and a wizard (of any stripe's) spell attack roll. Here's where it talks about a natural 20:
Though "doubling" is a bit misleading. Here's the critical hit mechanic:
The dice rolled are doubled (static mods are only counted once). Believe it or not this is only slightly more damaging on average than 4e's crit mechanic (though it is swingier). Let's look at a d8 in each mechanic. In 4e a d8 for damage would result in 8 damage on a crit. In 5e it would result in 2d8 rolled for an average of 9 (min of 2, max of 16). Obviously a good roll makes this a way swingier result, however, an average roll only makes this one point of damage better. This doesn't factor in the fact that in 4e you regularly add in your weapon's crit dice to your attacks, and this would lead to an average slightly above or about the same as 5e's crit damage.
Note: if you watch the D&D live streams they were talking about a fight against an Ogre with Fighter levels who kept critting them and dropping their PCs. (I think that's the right stream, it might have been the other Q&A).